Strong unique continuation for the Lamé system with Lipschitz coefficients in three dimensions

Abstract

Abstract. This paper studies the strong unique continuation property for the Lamé system of elasticity with variable Lamé coefficients λ, μ in three dimensions, div(μ(∇u + ∇u t )) + ∇(λdivu) + V u = 0 where λ and μ are Lipschitz continuous and V ∈ L ∞ . The method is based on the Carleman estimate with polynomial weights for the Lamé operator.Mathematics Subject Classification. 35B60, 74B05.

Stabilization of some elastodynamic systems with localized Kelvin-Voigt damping

“…• in [48], Ω is just a bounded domain in R 2 , the elasticity coefficients satisfy a ijkh ∈ W 1,∞ (Ω), and are further required to meet a special spectral condition (see (2.17) in [48]) • in [2], Ω is a bounded domain in R N , (N ≥ 2), and the material is isotropic: a ijkh (x) = 2µ(x)δ ik δ jh + λ(x)δ ij δ kh , where λ, µ ∈ C 1,1 (Ω) with µ(x) ≥ α 0 and 2µ(x) + λ(x) ≥ β 0 , for all x ∈ Ω, for some positive constants α 0 and β 0 . • in [70], Ω is a bounded domain in R 3 , (N = 3), and the material is isotropic as in the second point above, with λ and µ being Lipschitz continuous, and further satisfying the coerciveness conditions above.…”